Over a smooth closed manifold \(X\) let \(\mathcal A\) be an Azumaya bundle, a bundle of algebras with local trivializations associating the algebras with \(M_{m}(\mathbb{C})\). \(\mathcal A\) provides the structure needed to define twisted \(K\)-theory groups \(K^{0}(X,\mathcal A)\), which are generated by projective vector bundles. On a fiber bundle \(M\rightarrow X\) with smooth closed manifolds as fibers, the authors describe families of elliptic pseudodifferential operators acting on sections of projective vector bundles pulled back from \(X\). They show that the associated family of Fredholm operators (alternatively, the associated index difference bundle) defines an element of \(K^{0}(X,\mathcal A)\) known as the analytic index of the family, and they show that the symbols define a topological index in \(K^{0}(X,\mathcal A)\). The authors use an axiomatic approach to prove that the analytic index equals the topological index, iė. to generalize the Atiyah-Singer families index theorem to this setting. They also calculate the Chern character of the index and the Chern class of the associated determinant line bundle. Because of the paper’s dependence on finite-dimensional Azumaya algebras, the twisted \(K\)-theory groups treated are those asociated with torsion Dixmier-Douady invariants. String theory is one source of interest in twisted \(K\)-theory.